For the initial state \(|01\rangle \) , we check the entanglement between the two magnons by Eq. (REF ). We find that there is also no
entanglement. And we adapt the criteria proposed by Hillery and Zubairy [1]}. No entanglement has been found. Due to that \(|01\rangle \)  is non-Gaussian state, these judgement methods are sufficient, but not necessary, for the detection of entanglement. Whether there is entanglement is an open question, which is worth exploring. For the long evolution time, the state will approach the Gaussian state due to the considerably large number of particles and Gaussian noise. As a result, we can obtain that if there is entanglement, it should be small. For the closed PT symmetrical system as considered in Ref. [2]}, the pure Bell state (maximal entanglement state) can be obtained. However, for the open PT symmetrical system in this article, we realize that the quantum fluctuations will greatly reduce the degree of entanglement.
